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DKXD x>0 x khac 1
\(\frac{\sqrt{x}}{\sqrt{x}-1}-\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{x-1}{\sqrt{x}\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}+1}{\sqrt{x}}\)
a: \(P=\dfrac{x-\sqrt{x}-1-\sqrt{x}+1}{x-1}\cdot\dfrac{4\left(\sqrt{x}-2\right)}{\sqrt{x}\left(\sqrt{x}-2\right)^2}\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)\cdot4\left(\sqrt{x}-2\right)}{\sqrt{x}\left(x-1\right)}=\dfrac{4}{x-1}\)
Để P nguyên dương thì x-1 thuộc {1;4;2}
=>x thuộc {2;5;3}
b: x+y+z=0
=>x=-y-z; y=-x-z; z=-x-y
\(P=\dfrac{x^2}{y^2+z^2-\left(y+z\right)^2}+\dfrac{y^2}{z^2+x^2-\left(x+z\right)^2}+\dfrac{z^2}{x^2+y^2-\left(x+y\right)^2}\)
\(=\dfrac{x^2}{-2yz}+\dfrac{y^2}{-2xz}+\dfrac{z^2}{-2xy}\)
\(=\dfrac{x^3+y^3+z^3}{2xyz}\cdot\left(-1\right)\)
\(=-\dfrac{\left(x+y\right)^3+z^3-3xy\left(x+y\right)}{2xyz}\)
\(=-\dfrac{\left(-z\right)^3+z^3-3xy\cdot\left(-z\right)}{2xyz}=-\dfrac{3}{2}\)
đkxđ là \(x\ne1;x>0\)
\(Q=\frac{\sqrt{x}\left(\left(\sqrt{x}\right)^3-1\right)}{x+\sqrt{x}+1}-\frac{\sqrt{x}\left(2\sqrt{x}+1\right)}{\sqrt{x}}+\frac{2\left(x-1\right)}{\sqrt{x}-1}\)
\(Q=\frac{\sqrt{x}\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{x+\sqrt{x}+1}-2\sqrt{x}-1+\frac{2\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}-1}\)
\(Q=x-\sqrt{x}-2\sqrt{x}-1+2\sqrt{x}+2=x-\sqrt{x}+1\)
gtnn \(x-\sqrt{x}+1=x-\frac{1}{2}.2.\sqrt{x}+\frac{1}{4}+\frac{3}{4}=\left(\sqrt{x}-\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\)
gtnn 3/4
ý c bạn tự làm nha mk chịu
A = \(\frac{\left(x-\sqrt{x}\right)+x-1}{x-\sqrt{x}}\) = \(\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\)+ 1 = \(\frac{\sqrt{x}+1}{\sqrt{x}}\)+ 1 = 2 + \(\frac{1}{\sqrt{x}}\)
Do \(\sqrt{x}\)> 0 mà \(\frac{1}{\sqrt{x}}\)là số nguyên => \(\sqrt{x}\)= 1 => x = 1